Simplifying product of binomial coefficients

Simplifying product of binomial coefficients

Let $j,k,l,n$ be positive integers such that $j,k,l \leq n$, $j-l \leq k$
and $l \leq k$. Is there any way to simplify the product $$
\binom{n-j-k-l}{k-j-l}\binom{n-j-k+l}{l}^2, $$ perhaps as a product of
just two binomial coefficients (or just as one such coefficient)? I've
been staring at this all day and can't make any sense of it.
These integers appear while calculating the square of a norm of a
$(k,k)$-form on a hermitian vector space of dimension $n$; we express the
square of the norm as a linear combination of various products of $l$-th
powers of the adjoint of the Lefschetz operator applied to the $j$-th
elements in the primitive decomposition of the form (times a power of the
exterior form that the inner product defines). The norm we want is a
linear combination of some things and the coefficients of that linear
combination are these monstrous binomial coefficients, so I'd be very
happy if they simplify a little bit.